The Republican senator encouraged the former Democratic vice president to “not walk away” from politics, as Mr. Biden put it before refusing to discuss a possible 2020 presidential run. Mr. McCain is using a new book and documentary to reveal his regret about not selecting former Senator Joseph I. Lieberman as his running mate in 2008. His intimates have informed the White House that their current plan for his funeral is for Vice President Mike Pence to attend the service to be held in Washington’s National Cathedral but not President Trump, with whom Mr. McCain has had a rocky relationship.

And some of his associates, though not his family, have started to quietly put out word that they want a “McCain person” eventually appointed to fill his Senate seat, a roster that includes his wife, Cindy.